Global moduli spaces and simple holomorphic bundles
type: Item
instance of: scholarly article
title: Global moduli spaces and simple holomorphic bundles
zbMATH Open document ID: 0679.32024
author: Siegmund Kosarew
author: Christian Okonek
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1989-01-01T00:00:00Z
review text: The authors prove the following theorem: Let S be a complex space and F a groupoid defined on the category (An/S) of complex spaces over S with values in the category of sets. Let \(| F|\) denote the associated isomorphism class functor and let \(k(s)\cong {\mathbb{C}}\) be te residue field of \(s\in S\). Assume that: (1) Each object of \(| F| (Spec(k(s)))\), \(s\in S\), possesses a versal deformation and versality is an open property. (2) For all objects of \(a,b\in F(T)\), \(T\in (An/S)\), the isomorphism functor \(Isom^ F_ T(a,b)\) is representable by a separated morphism \(Z(a,b)\to T\) such that \(Z(a,a)\) is smooth over T and the operation of \(Z(a,a)\) on \(Z(a,b)\) is locally proper (resp. proper). Then \(| F|\) has a coarse moduli-space (resp. one with a separated structure map). The proof consists of a local construction and a gluing procedure and shows actually a bit more. Namely that the sheafified functor associated to \(| F|\) is representable. The theorem is then applied to \(F=Coh_ S(X/S)\) (where \(X\to S\) is a fixed separated map of complex spaces) which associates to \(T\in (An/S)\) the set of coherent \({\mathcal O}_{X_ T}\)-modules F, flat and proper over S, which are simple (i.e. \(T\times {\mathbb{C}}\to End_ X(F)\), \((t,\lambda)\mapsto \lambda id_{F(t)}\), is an isomorphism of linear fibre spaces over T). The result is that the functor \(| Coh_ S(X/S)|\) of isomorphism classes of simple coherent sheaves on X/S has a coarse moduli space. In particular, simple coherent sheaves and simple holomorphic bundles on a fixed compact complex space have a coarse moduli space. Note that these moduli spaces are in general not separated. It should be mentioned that if \(X\to S\) is a locally projective, finitely presented morphism of schemes, then \(| Coh_ S(X/S)|\) has a coarse moduli space which is a quasi-separated algebraic space in the sense of M. Artin. This was proved before by \textit{A. B. Altman} and \textit{S. L. Kleiman} [Adv. Math. 35, 50-11…
reviewed by: Gert-Martin Greuel
Mathematics Subject Classification ID: 32G13
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 4113002
zbMATH Keywords: moduli space of simple holomorphic bundles
zbMATH Keywords: coarse moduli space of analytic functors
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195173759
OpenAlex ID: W1980248668
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DOI: 10.2977/PRIMS/1195173759
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description: scientific article; zbMATH DE number 4113002